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Graph-Theoretic Representations for Proximity Matrices Through Strongly-Anti-Robinson or Circular Strongly-Anti-Robinson Matrices

Published online by Cambridge University Press:  01 January 2025

Lawrence Hubert ,
Phipps Arabie  and
Jacqueline Meulman
Lawrence Hubert*
Affiliation:
University of Illinois, Champaign
Phipps Arabie
Affiliation:
Faculty of Management, Rutgers University, Newark, New Jersey
Jacqueline Meulman
Affiliation:
Department of Data Theory, Leiden University
*
Requests for reprints should be sent to Lawrence Hubert, Department of Psychology, The University of Illinois, 603 East Daniel Street, Champaign, IL 61820, USA.
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Abstract

There are various optimization strategies for approximating, through the minimization of a least-squares loss function, a given symmetric proximity matrix by a sum of matrices each subject to some collection of order constraints on its entries. We extend these approaches to include components in the approximating sum that satisfy what are called the strongly-anti-Robinson (SAR) or circular strongly-anti-Robinson (CSAR) restrictions. A matrix that is SAR or CSAR is representable by a particular graph-theoretic structure, where each matrix entry is reproducible from certain minimum path lengths in the graph. One published proximity matrix is used extensively to illustrate the types of approximation that ensue when the SAR or CSAR constraints are imposed.

Keywords

least-squares matrix approximationgraphical representationstrongly-anti-Robinsoncircular strongly-anti-Robinson
Type
Original Paper
Information
Psychometrika , Volume 63 , Issue 4 , December 1998 , pp. 341 - 358
Copyright
Copyright © 1998 The Psychometric Society

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Footnotes

The authors are indebted to Boris Mirkin who first noted in a personal communication to one of us (LH, April 22, 1996) that the optimization method for fitting anti-Robinson matrices in Hubert and Arabie (1994) should be extendable to the fitting of strongly anti-Robinson matrices as well.

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